A Switch Criterion for Defined Contribution Pension Schemes

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A Switch Criterion for Defined Contribution
Pension Schemes

• Bas Arts and Elena Vigna

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Basic Idea

• Investing the contributions into equities a
  certain period and then wait for the right time
  to switch into bonds
• Inspired by
• Mean Reversion of Equities
• Lifestyle followed by Income Drawdown leads to
  discontinuity in portfolio composition
• The idea of extra saving or reserve required in
  DC schemes
• Considering both the accumulation and the
  distribution phase

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Assumptions and Target Return

• 2-assets equities and bonds with lognormal
  distribution
• Equities, real force of interest ?t ? N(?, ?22),
  IID
• Bonds, real force of interest ?t ? N(?, ?12), IID
  
• ?t and ?t are uncorrelated
• c, contribution rate (constant)
• Target Return
• (Chisini Average)

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The basic strategy

• The aim is to minimize the probability of failing
  the target pension
• Find the optimal number of years for investing
  the contributions into equities SC
• After SC the new contributions are invested into
  bonds, while the old contributions remain
  invested into equities (equity fund)
• Propose an optimal criterion for switching the
  equity fund from equities into bonds (SF) using a
  dynamic approach

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6 important moments in life

• I Moment of joining the scheme
• SC Switch of the contributions
• SF Switch of the equity fund (maybe)
• R Age of retirement
• A Age when annutization is compulsory
• D Death

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Timeline
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Initial SC and SF

• Looking only at expected returns, we calculate
  the switch of contributions (SC) as follows

TargetFund
Bond Fund
Equity Fund
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Example

• The following parameter values have been chosen
• µ4 ?6 sµ5 s ? 15
• With a contribution of c1 and 40 years to
  retirement, this results in a Target Return of
  5.3125 and in a Target Fund ( ) of
  142,50 at retirement and the initial switch of
  the contributions from equities to bonds (SC)
  will be 23

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SC for different Target Returns
Further research Sensitivity Analysis to take
into account risk aversion
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Dynamic Switch Criterion

• From time tSC on at the beginning of each year
  we check whether the projected future value fund
  of the realized fund at time t together
  with future contributions is greater than or
  equal to the Target Fund
• If this is the case then the equity fund will be
  converted in bonds otherwise it remains invested
  into equities for at least one more year, while
  investing the new contributions in bonds
• In formula the SF occurs at the first time the
  following holds

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Figure 1High return on equities
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Figure 2 Lower than expected return on equities
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Figure 3 Low return on equities
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Comparison other strategies
(TARGET FUND 142,50)
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Comments

• The mean of the Switch Strategy is much lower
  than the mean of the other strategies and at the
  same time the probability of failing the target
  fund is higher
• The higher standard deviation of the other
  strategies can for a great part be explained by
  the surplus of the final fund on the Target Fund
  ( the other risk measures are comparable)

This is because the current Switch Criterion
ignores the fact that bonds have their risk as
well
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Adjustments to Basic Strategy

• Investing the contributions some extra years in
  equities affects SC ( SF)
• Including a reserve when calculating SF
• affects SF

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Flexible SC
Comments the difference between the average fund
at t and the target fund increases with time
because the fund remains invested longer in
equities the probability that SC and SF coincide
increases the probability of failing the target
remarkably decreases when SC increases
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Comparison other Strategies
Switch Strategy
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Comments

• Comparing the SC31 with SC23 strategy
• the mean is higher while the probability of
  failing the Target Fund is lower
• the standard deviation, the downside deviation
  and the mean shortfall are slightly higher (but
  considering the 36 lowest values - in case of
  failure - of SC23 these risk measures are very
  similar)
• in the worst cases (VaR95) the final fund is
  lower for SC31 while the VaR75 is higher

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Reserve at time t

• Estimating the mean shortfall of the final fund
  for each year t (SMS), given that the yearly
  target (YT) is exactly satisfied at time t.
• Simulated future Fund
• New Criterion
• With

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Linear Regression Reserve
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Adjusted Switch strategy in comparison with other
strategies
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Comments

• The mean in comparison with the other strategies
  remains lower but the probability of failing the
  Target Fund is lower as well
• The VaR95 is lower than the Var95 of the
  lifestyle strategy, while the VaR75 is higher
  than in both the other strategies
• The probability of failing the Target Fund, given
  that the SF occurred, is only 4. This is
  important for the Income Drawdown option in the
  distribution phase

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Distribution Phase

• The Criterion changes Income Drawdown (only if
  the switch of the equity fund SF didnt occur)
• For the pension P we take the pension that would
  have been obtained with the Target Fund
• While the fund in bonds gt 0 the pension will be
  deducted from this fund else it will be deducted
  from the equity fund
• We include a bonus factor for pooling
• The pensioner annuitizes at age A or before if
  the fund is big enough to buy the target pension,
  i.e. when

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Results Distribution Phase
P(No Drawdown
)

Drawdown/initial people


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Comments

• Total probability of failing the Target Fund for
  the Switch Strategy now becomes
  16,34,92,824 and 23,4 for the 100
  equity strategy
• The 100 equity strategy has a higher probability
  of reaching the desired pension than the other
  strategies if we take into consideration the
  income drawdown option
• The average of the fund in the cases where the
  Target is not reached is higher for the Switch
  Strategy and the SF occurs on average earlier
• Income drawdown for the lifestyle strategy has
  not been done, because the fund is fully invested
  in bonds at retirement

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Conclusion

• The adjusted Switch Strategy seems to be suitable
  for DC schemes for the following reasons
• it allows for a first partial switch of the fund
  from equities into bonds (in order to limit the
  risk), but considers also actual returns from the
  financial market through a dynamic criterion for
  the second and definitive switch
• numerical results are good in comparison with
  other investment strategies for DC schemes
• it considers both the accumulation and the
  distribution phase so that discontinuity in
  portfolio composition when applying income
  drawdonw (like lifestyle) is avoided
• Furthermore, investing fully in equities seems to
  be less risky than usually considered

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Further Research

• Finding a more appropriate estimate for the
  reserve
• Introduce deferred annuities as a third
  investment possibility
• Taking into account the current yield on bonds
  at any time t, instead of considering a constant
  expected return

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